Mathematical logic and foundations Definition

Universal quantifier

Definition

Let $X$ be a set and let $P(x)$ be a predicate on $X$ . The universal quantification of $P$ over $X$ is the proposition $\forall x \in X, \, P(x)$ , read "for all $x$ in $X$ , $P(x)$ ". It is true when $P(a)$ is true for each $a \in X$ , and false otherwise.

Remarks

When $X$ is empty, $\forall x \in X, \, P(x)$ is true for each $P$ . This is a case of vacuous truth. For a finite set $X = \{a_1, \dots, a_n\}$ , the statement is equivalent to $P(a_1) \land \dots \land P(a_n)$ .

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Proposition concept
Predicate definition
Universal quantifier definition

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Tag	`0B88PQTJ66`
URI	`m2://0B88PQTJ66`
Kind	definition
Writing	W1 stated
MSC	03B10
Defines	Universal quantifier
Uses	Predicate

Universal quantifier

Definition

Remarks

Reading path

Where this leads

Further reading

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